maximal subgroup

Let G be a group.

A subgroupH of G
is said to be a maximal subgroup of G
if H≠G and there is no subgroup K of G
such that H<K<G.
Note that a maximal subgroup of G is not maximal (http://planetmath.org/MaximalElement) among all subgroups of G,
but only among all proper subgroups of G.
For this reason, maximal subgroups are sometimes called maximal proper subgroups.

Similarly, a normal subgroupN of G
is said to be a maximal normal subgroup
(or maximal proper normal subgroup) of G
if N≠G and there is no normal subgroup K of G
such that N<K<G.
We have the following theorem:

Theorem.

A normal subgroup N of a group G is a maximal normal subgroup
if and only if the quotient (http://planetmath.org/QuotientGroup) G/N
is simple (http://planetmath.org/Simple).